Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbfv2g Structured version   Visualization version   GIF version

Theorem csbfv2g 6142
 Description: Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbfv2g (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐹𝐴 / 𝑥𝐵))
Distinct variable group:   𝑥,𝐹
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem csbfv2g
StepHypRef Expression
1 csbfv12 6141 . 2 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
2 csbconstg 3512 . . 3 (𝐴𝐶𝐴 / 𝑥𝐹 = 𝐹)
32fveq1d 6105 . 2 (𝐴𝐶 → (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (𝐹𝐴 / 𝑥𝐵))
41, 3syl5eq 2656 1 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐹𝐴 / 𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ⦋csb 3499  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717  ax-pow 4769 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812 This theorem is referenced by:  csbfv  6143  ixpsnval  7797  swrdspsleq  13301  sumeq2ii  14271  fsumabs  14374  prodeq2ii  14482  fprodabs  14543  ixpsnbasval  19030  coe1fzgsumdlem  19492  evl1gsumdlem  19541  pm2mp  20449  cayhamlem4  20512  nbgraopALT  25953  iuninc  28761  cdlemk39s  35245
 Copyright terms: Public domain W3C validator